I am planning to move out of my rental home on the 15th of the month. Does the landlord by law have to Prorate my last months rent or is up to the landlord?

It is up to the rental agreement, most notice is from pay period to pay period, so if you wanted to move out on the 15th of next month you should have given a 45 day noice last month on the 1st for next month.
If you give a 30 day notice on the 15th, in most rental agreements you are liable for the full month not just 1/2 of the month. So no prorating.
Now if the landlord will do it, great, I used to but don't expect it from most.